Geometry
Ed Angel
Professor of Computer Science,
Electrical and Computer
Engineering, and Media Arts
University of New Mexico
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
1
Objectives
• Introduce the elements of geometry
- Scalars
- Vectors
- Points
• Develop mathematical operations among
them in a coordinate-free manner
• Define basic primitives
- Line segments
- Polygons
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
2
Basic Elements
• Geometry is the study of the relationships
among objects in an n-dimensional space
- In computer graphics, we are interested in
objects that exist in three dimensions
• Want a minimum set of primitives from which we
can build more sophisticated objects
• We will need three basic elements
- Scalars
- Vectors
- Points
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
3
Coordinate-Free Geometry
• When we learned simple geometry, most of us started
with a Cartesian approach
- Points were at locations in space p=(x,y,z)
- We derived results by algebraic manipulations
involving these coordinates
• This approach was nonphysical
- Physically, points exist regardless of the location of
an arbitrary coordinate system
- Most geometric results are independent of the
coordinate system
- Example Euclidean geometry: two triangles are
identical if two corresponding sides and the angle
between them are identical
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
4
Scalars
• Need three basic elements in geometry
- Scalars, Vectors, Points
• Scalars can be defined as members of sets
which can be combined by two operations
(addition and multiplication) obeying some
fundamental axioms (associativity, commutivity,
inverses)
• Examples include the real and complex number
systems under the ordinary rules with which we
are familiar
• Scalars alone have no geometric properties
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
5
Vectors
• Physical definition: a vector is a quantity
with two attributes
- Direction
- Magnitude
• Examples include
- Force
- Velocity
- Directed line segments
• Most important example for graphics
• Can map to other types
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
v
6
Vector Operations
• Every vector has an inverse
- Same magnitude but points in opposite direction
• Every vector can be multiplied by a scalar
• There is a zero vector
- Zero magnitude, undefined orientation
• The sum of any two vectors is a vector
- Use head-to-tail axiom
v
-v
v
w
v
u
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
7
Linear Vector Spaces
• Mathematical system for manipulating vectors
• Operations
- Scalar-vector multiplication u=v
- Vector-vector addition: w=u+v
• Expressions such as
v=u+2w-3r
Make sense in a vector space
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
8
Vectors Lack Position
• These vectors are identical
- Same length and magnitude
• Vectors spaces insufficient for geometry
- Need points
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
9
Points
• Location in space
• Operations allowed between points and
vectors
- Point-point subtraction yields a vector
- Equivalent to point-vector addition
v=P-Q
P=v+Q
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
10
Affine Spaces
• Point + a vector space
• Operations
- Vector-vector addition
- Scalar-vector multiplication
- Point-vector addition
- Scalar-scalar operations
• For any point define
-1•P=P
- 0 • P = 0 (zero vector)
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
11
Lines
• Consider all points of the form
- P()=P0 +  d
- Set of all points that pass through P0 in the
direction of the vector d
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
12
Parametric Form
• This form is known as the parametric form
of the line
- More robust and general than other forms
- Extends to curves and surfaces
• Two-dimensional forms
- Explicit: y = mx +h
- Implicit: ax + by +c =0
- Parametric:
x() = x0 + (1-)x1
y() = y0 + (1-)y1
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
13
Rays and Line Segments
• If  >= 0, then P() is the ray leaving P0 in
the direction d
If we use two points to define v, then
P( ) = Q +  (R-Q)=Q+v
=R + (1-)Q
For 0<=<=1 we get all the
points on the line segment
joining R and Q
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
14
Convexity
• An object is convex iff for any two points
in the object all points on the line segment
between these points are also in the
object
P
P
Q
convex
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
Q
not convex
15
Affine Sums
• Consider the “sum”
P=1P1+2P2+…..+nPn
Can show by induction that this sum makes
sense iff
1+2+…..n=1
in which case we have the affine sum of
the points P1,P2,…..Pn
• If, in addition, i>=0, we have the convex
hull of P1,P2,…..Pn
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
16
Convex Hull
• Smallest convex object containing P1,P2,…..Pn
• Formed by “shrink wrapping” points
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
17
Curves and Surfaces
• Curves are one parameter entities of the
form P() where the function is nonlinear
• Surfaces are formed from two-parameter
functions P(, b)
- Linear functions give planes and polygons
P()
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
P(, b)
18
Planes
• A plane can be defined by a point and two
vectors or by three points
P
v
Q
R
u
P(,b)=R+u+bv
R
P(,b)=R+(Q-R)+b(P-Q)
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
19
Triangles
convex sum of S() and R
convex sum of P and Q
for 0<=,b<=1, we get all points in triangle
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
20
Normals
• Every plane has a vector n normal
(perpendicular, orthogonal) to it
• From point-two vector form P(,b)=R+u+bv, we
know we can use the cross product to find
n = u  v and the equivalent form
(P()-P)  n=0
v
P
Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005
u
21